ATLAS: Unitary group U₅(3)

Order = 258190571520 = 2¹¹.3¹⁰.5.7.61.
Mult = 1.
Out = 2.

The page for the group 3¹⁰.U₅(3) (nonsplit extension) is available here.

The following information is available for U₅(3):

Standard generators.
Automorphisms.
Presentations.
Representations.
Maximal subgroups.

Standard generators

Standard generators for U₅(3) are a and b where a is in class 3A, b has order 5 and ab has order 16.
Standard generators for U₅(3):2 are c and d where c is in class ??, d has order ? and cd has order ??.

NB: Class 3A is the class of transvections in U₅(3).

Automorphisms

An [outer] automorphism of U₅(3) of order 2 can be obtained by mapping (a, b) to (a^-1, b).

Presentations

< a, b | a³ = b⁵ = [a, bab^-1ab] = [a, b^-2ab²] = (babababa^-1)⁴ = (ababa^-1bab²)⁵ = 1 >.

Representations

The representations of U₅(3) available are:

Dimension 5[a] over GF(9) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 5[b] over GF(9) - dual of the above: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 10[a] over GF(9) - skew square of 5a: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 10[b] over GF(9) - skew square of 5b: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 10 over GF(3) - reducible over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 15[a] over GF(9) - symmetric square of 5a: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 15[b] over GF(9) - symmetric square of 5b: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 20 over GF(3) - reducible over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 24 over GF(3) - adjoint representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 30[a] over GF(9) - in 5a × 10a: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 30[b] over GF(9) - in 5b × 10b: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 30 over GF(3) - reducible over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 51 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

Maximal subgroups

The maximal subgroups of U₅(3) are (I reckon) as follows [implementation of word programs not checked]:

3¹⁺⁶:(U₃(3) × 8), with generators ??, ??.
4.U₄(3).4, with generators a, bab^3.
3⁴⁺⁴:2.(A₆ × 4).2.
U₃(3) × 4.S₄.
U₄(2):2, with generators ??, ??.
4⁴.S₅, with generators ??, ??.
61:5 = F₃₀₅.

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